What you describe is not a problem of the burners, but of the furnace. You can take out only from the output stacks. That's much easier with burners: They have only one stack, which can be used as input AND output.

I think this suggestion is not possible, cause it changes basic game-play. But ...

Summary: I think burner insterters should be able to pull coal to fuel themselves out of furnaces.

Reasoning: Burner Inserters have a limited number applications where they are useful or efficient. The primary one is in very dense steam power setups, however most players switch to solar as soon as ...